I'm confused as to how to develop intuition for the "classical" results of topology. By these, I mean results like Brouwer's fixed point theorem, the hairy-ball theorem, Borsuk-Ulam theorem, fundamental theorem of algebra, etc., things that can be proven with knowledge of the fundamental group of the circle.
I'm currently reading Chapter 9 of Munkres' Topology, and I can follow the arguments presented in the text. However, I don't feel as though I'm understanding them intuitively. Is there a reference that could act as a supplement to this chapter that treats these "classic" theorems in more depth with emphasis on intuitive understanding? How did you guys develop this intuition and begin "thinking like a topologist"?